Comment by aleph_minus_one

A¹ homotopy is currently still a research area, but if you read the linked Wikipedia article:

"The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line A¹, which is."

In other words: it's a novel approach towards homotopy theory, which does have applications in the physical world.

Even better (Wikipedia article):

"[A¹ homotopy theory] has also recently revolutionized the theory of enumerative geometry problems."

Enumerative geometry does have applications in the physical world.